Today I cookerized two rabbits (you could use chicken) for an hour (on manual) in chicken broth, then shredded the meat.  I saved half to freeze for a future meal and used the chicken/rabbit broth to cook some garlic butter noodles (below). To the meat, I added BBQ sauce, honey, salt, and pepper.  (I usually use leftover BBQ sauce from whatever meat I made last in the Instant Pot.)  Served these on giant buns.
For the noodles, I used a box of Rigatoni and added enough broth to cover the noodles, some butter, garlic, and oregano. 4 minutes on manual. The noodles were great, but I had to drain and add more butter, oregano, and garlic salt for more flavor. They were a great side dish and a good reuse of the broth.
Instant Pot pulled rabbit BBQ sandwiches could of course easily be pulled chicken BBQ sandwiches. We make them with rabbit, chicken, squirrel, deer, whatever's handy.